Asymptotic dimension of one relator groups

TL;DR

This paper demonstrates that one relator groups have finite asymptotic dimension when viewed as metric spaces, providing an estimate based on the length of the relator, advancing understanding of their large-scale geometry.

Contribution

It establishes the finiteness of asymptotic dimension for one relator groups and offers an explicit estimate related to relator length, a novel result in geometric group theory.

Findings

One relator groups have finite asymptotic dimension.

An explicit estimate of asymptotic dimension based on relator length.

Provides new insights into the large-scale geometry of one relator groups.

Abstract

We show that one relator groups viewed as metric spaces with respect to the word-length metric have finite asymptotic dimension in the sense of Gromov and give an estimate of their asymptotic dimension in terms of the relator length.